In ordinary universal algebra we expect to be able to form limits at the level of underlying sets, which are then given a suitable algebra structure. When one considers algebraic structures borne not by sets but by categories, typically the morphisms preserve structure only in some weak sense. When one uses these weak morphisms, it is no longer the case that limits can be formed as in the previous paragraph. In what I expect to be a series of (about) two talks, I will talk about existing approaches to this problem and their strengths and weaknesses, as well as a new setting being developed in joint work with Mike Shulman.